commutants of finite blaschke product multiplication ...ccowen/talks/wabash1405.pdfcommutants of...

Commutants of Finite Blaschke Product

Multiplication Operators on Bergman Spaces

Carl C. Cowen

IUPUI

(Indiana University Purdue University Indianapolis)

Wabash Modern Analysis Seminar, April 12, 2014

Commutants of Finite Blaschke Product

Multiplication Operators on Bergman Spaces

Carl C. Cowen

IUPUI

(Indiana University Purdue University Indianapolis)

Wabash Modern Analysis Seminar, April 12, 2014

joint work with Rebecca Wahl (Butler University)

In this talk H will denote a Hilbert space of analytic functions on D,

Usual spaces: f analytic in D, with f (z) =∑∞

n=0 anzn

Hardy: H2(D) = H2 = {f : ‖f‖2 =

∞∑n=0

|an|2 <∞}

Bergman: A2(D) = A2 = {f : ‖f‖2 =

∫D|f (z)|2 dA(z)

π<∞}

weighted Bergman (γ > −1): A2γ = {f : ‖f‖2 =

∫D|f (z)|2(1−|z|2)γ dA(z)

π<∞}

weighted Hardy (‖zn‖ = ωn > 0): H2(ω) = {f : ‖f‖2 =

∞∑n=0

|an|2ω2n <∞}

In these spaces, for α in D, the linear functionals f 7→ f (α) are bounded.


In Hilbert space, these linear functionals are given by the inner product:

the reproducing kernel function for H is Kα in H with

〈f,Kα〉 = f (α) for all f ∈ H


In Hilbert space, these linear functionals are given by the inner product:

the reproducing kernel function for H is Kα in H with

〈f,Kα〉 = f (α) for all f ∈ H

For H2, we have Kα(z) = (1− αz)−1

For A2, we have Kα(z) = (1− αz)−2

In this talk, we will consider spaces H2κ for κ ≥ 1 which are the

weighted Hardy spaces with

Kα(z) = (1− αz)−κ

The spaces H2κ include the usual Hardy and Bergman spaces and

all the weighted Bergman spaces (γ = κ + 2).

Conversation with Axler made it clear that right generality is to consider

Hilbert spaces, H, of functions analytic on D that satisfy:

(I) The constant function 1(z) ≡ 1 for z in D is in H and ‖1‖ = 1

(II) For α in D, the linear functional f 7→ f (α) is continuous on H

(III) For ψ in H∞, operator Tψ given by (Tψf )(z) = ψ(z)f (z) is in B(H).

(IV) For α in D and f in H with f (α) = 0, then f/(z − α) is also in H.







• Conditions (I) & (III) say H and its multiplier algebra contain H∞








• Condition (II) says H has kernel functions & its multiplier algebra is H∞








• Condition (II) says H has kernel functions & its multiplier algebra is H∞

• For ψ in H∞, the operator Tψ in condition (III) is called

an analytic multiplication operator or an analytic Toeplitz operator

and conditions imply ‖Tψ‖ = ‖ψ‖∞ and this means ‖ψ‖ ≤ ‖ψ‖∞

The Hardy space H2, the Bergman space A2, and the standard weight

Bergman spaces H2κ satisfy Conditions (I), (II), (III), and (IV).

The usual Dirichlet space, and many weighted Dirichlet spaces, do not

satisfy all the conditions: not all H∞ functions are in Dirichlet space!



Consequence: if f is in H, ψ is bounded analytic function, and α is in D,

〈f, T∗ψKα〉 = 〈Tψf,Kα〉 = ψ(α)f (α) = ψ(α)〈f,Kα〉 = 〈f, ψ(α)Kα〉



Consequence: if f is in H, ψ is bounded analytic function, and α is in D,

〈f, T∗ψKα〉 = 〈Tψf,Kα〉 = ψ(α)f (α) = ψ(α)〈f,Kα〉 = 〈f, ψ(α)Kα〉

Since f is arbitrary, this means T∗ψKα = ψ(α)Kα

and every kernel function is an eigenvector for T∗ψ .

The spectrum of Tψ is the closure of ψ(D), there no eigenvalues for Tψ,

but the complex conjugate of ψ(D) consists of eigenvalues of T∗ψ .

Definition:

An inner function is a bounded analytic function, ψ, on D such that

limr→1−

|ψ(reiθ)| = 1 a. e. dθ

Definition:


limr→1−


Definition:

A function B is a Blaschke product of order n if it can be written as

B(z) = µ

(ζ1 − z1− ζ1z

)(ζ2 − z1− ζ2z

)· · ·(ζn − z1− ζnz

)where |µ| = 1 and ζ1, ζ2, · · ·, ζn are points of D.

Blaschke products of order n are inner functions

and map the closed disk n - to - 1 onto itself.

Definition:


limr→1−


Definition:

A function B is a Blaschke product of order n if it can be written as

B(z) = µ

(ζ1 − z1− ζ1z

)(ζ2 − z1− ζ2z

)· · ·(ζn − z1− ζnz

)where |µ| = 1 and ζ1, ζ2, · · ·, ζn are points of D.

Blaschke products of order n are inner functions

and map the closed disk n - to - 1 onto itself.

For ψ, a non-constant inner function, the multiplication operator Tψ is a

pure isometry on H2 but is not isometric on the Bergman spaces.

Beurling’s Theorem (1949):

Let Tz be the operator of multiplication by z on H2(D). A closed subspace

M of H2(D) is invariant for Tz if and only if there is an inner function ψ

such that M = ψH2(D).

This result is indicative of the interest in the operator Tz of multiplication

by z on H2(D) and in analytic Toeplitz operators Tψ on Hilbert spaces of

analytic functions more generally.

Definition:

If A is a bounded operator on a space H, the commutant of A is the set

{A}′ = {S ∈ B(H) : AS = SA}

For example, for Tz on H2,

{Tz}′ = {Tψ : ψ ∈ H∞}0 0 0 · · ·

1 0 0 · · ·

0 1 0 . . .

a00 a01 a02 · · ·

a10 a11 a12 · · ·

a20 a21 a22 . . .

=

0 0 0 · · ·

a00 a01 a02 · · ·

a10 a11 a12 . . .

Definition:


{A}′ = {S ∈ B(H) : AS = SA}

For example, for Tz on H2,

{Tz}′ = {Tψ : ψ ∈ H∞}0 0 0 · · ·

1 0 0 · · ·

0 1 0 . . .

a00 a01 a02 · · ·

a10 a11 a12 · · ·

a20 a21 a22 . . .

=

0 0 0 · · ·

a00 a01 a02 · · ·

a10 a11 a12 . . .

a00 a01 a02 · · ·

a10 a11 a12 · · ·

a20 a21 a22 . . .

0 0 0 · · ·

1 0 0 · · ·

0 1 0 . . .

=

a01 a02 a03 · · ·

a11 a12 a13 · · ·

a21 a22 a23 . . .

0 0 0 · · ·

a00 a01 a02 · · ·

a10 a11 a12 . . .

=

a01 a02 a03 · · ·

a11 a12 a13 · · ·

a21 a22 a23 . . .

This means that a0j = 0 for j ≥ 1 and ai,j = ai+1,j+1 for i, j ≥ 0

In particular, the matrix is lower triangular and is constant along diagonals:

a0 0 0 0 · · ·

a1 a0 0 0 · · ·

a2 a1 a0 0 · · ·

a3 a2 a1 a0... ... ... . . .

This is Tψ for ψ(z) =

∑∞j=0 ajz

j where ‖ψ‖∞ = ‖Tψ‖.

Definition:


{A}′ = {S ∈ B(H) : AS = SA}

We have seen for Tz on H2,

{Tz}′ = {Tψ : ψ ∈ H∞}

By the 1970’s, there was interest in the more general question,

For ψ in H∞ and Tψ an operator on H2, what is {Tψ}′ ?

or more specifically,

For B a finite Blaschke product and TB operating on H2, what is {TB}′ ?

Deddens & Wong’s 1973 paper used the fact that for B a finite Blaschke

product, the operator TB acting on H2 is a pure isometry to show that

The operator S in B(H2) is in {TB}′ if and only if

S can be represented as a lower triangular block Toeplitz matrix

with respect to the description of H2 as⊕∞

k=0BkW where W is the

wandering subspace W = (BH2)⊥, that is,

S =

A0 0 0 0 · · ·

A1 A0 0 0 · · ·

A2 A1 A0 0 · · ·

A3 A2 A1 A0 · · ·... . . .

Shortly thereafter, Thomson’s papers and Cowen’s papers computed {TB}′

from a different perspective:

Fundamental Lemma:

For S a bounded operator on H2 and ψ in H∞, these • are equivalent

• S commutes with Tψ

• For all α in D, S∗Kα ⊥ (ψ − ψ(α))H2

Proof: (Main calculation)

For α in D, ψ in H∞, and STψ = TψS, if f is in H2,

〈(ψ−ψ(α))f, S∗Kα〉 = 〈STψf,Kα〉−ψ(α)〈Sf,Kα〉

= 〈TψSf,Kα〉 − ψ(α)〈Sf,Kα〉 = 〈Sf, T∗ψKα〉 − ψ(α)〈Sf,Kα〉

= ψ(α)(Sf )(α)− ψ(α)(Sf )(α) = 0

The main results of these papers were to identify some special classes of

bounded analytic functions whose Toeplitz operators have commutants that

exemplify the possible commutants of analytic Toeplitz operators.

That is, maybe there is a small set S of H∞ functions so that for each ψ in

H∞, there is ϕ in S so that {Tψ}′ = {Tϕ}′.




That is, maybe there is a small set S of H∞ functions so that for each ψ in

H∞, there is ϕ in S so that {Tψ}′ = {Tϕ}′.

It became clear that, inner functions and covering maps should be part of

any such set S because Toeplitz operators associated with many other H∞

functions have commutants the same as inner function or covering map

Toeplitz operators.




For example, the Fundamental Lemma, immediately implies

If ϕ and ψ are in H∞ and there is an analytic function g

so that ϕ = g ◦ ψ, then {Tϕ}′ ⊃ {Tψ}′.

So a natural question is: “If ϕ = g ◦ ψ, when does {Tϕ}′ = {Tψ}′ ?”




Theorem: [C., 1978]

If ψ is a bounded analytic function on the disk D

and α0 is a point of the disk so that the inner factor of ψ − ψ(α0)

is a finite Blaschke product,

then there is a finite Blaschke product B so that

{Tψ}′ = {TB}′




Theorem: [C., 1978]



is a finite Blaschke product,

then there is a finite Blaschke product B so that

{Tψ}′ = {TB}′

In fact, the Blaschke product B is the “largest” inner function for which

there is bounded function g so that ψ = g ◦B.

For B a finite Blaschke product of order n, except for n(n− 1) points of the

disk for which B(α) = B(β) and B′(β) = 0,(((B −B(α))H2

)⊥= span {Kβ1, Kβ2, · · · , Kβn}

where the points α = β1, β2, · · ·, βn are the n distinct points of D for which

B(βj) = B(α).


disk for which B(α) = B(β) and B′(β) = 0,((B −B(α))H2

)⊥= span {Kβ1, Kβ2, · · · , Kβn}


B(βj) = B(α).

The important fact behind this work is that the kernel functions Kα,

Kα(z) = (1− αz)−1 in H2 and Kα(z) = (1− αz)−2 in A2, depend

conjugate analytically on α, so if A is a linear operator so that AKα is

always in((B −B(α))H2

)⊥, then

AKα =∑j

cjKβj

where the cj’s and the Kβj ’s are conjugate analytic in α


disk for which B(α) = B(β) and B′(β) = 0,(((B −B(α))H2

)⊥= span {Kβ1, Kβ2, · · · , Kβn}


B(βj) = B(α).

Observation:

For the study of commutants of Toeplitz operators, it is more important

that a Blaschke product B is an n - to -1 map of D onto itself than the fact

that TB is a pure isometry on H2.

Of course, since the points α = β1, β2, · · ·, βn depend on α, we may write

them as α = β1(α), β2(α), · · ·, βn(α).

In fact (!), if B is a finite Blaschke product of order n and α is a point of

the disk that is NOT one of the n(n− 1) points of the disk for which

B(α) = B(β) and B′(β) = 0,

the maps α 7→ βj(α) are just the n branches of the analytic function

B−1 ◦B that is defined and arbitrarily continuable on the disk with the

n(n− 1) exceptional points removed.

Of course, since the points α = β1, β2, · · ·, βn depend on α, we may write

them as α = β1(α), β2(α), · · ·, βn(α).



B(α) = B(β) and B′(β) = 0,




Theorem: (Cowen, 1974)

For B a finite Blaschke product, the branches of B−1 ◦B form a group

whose normal subgroups are associated with compositional

factorizations of B into compositions of two Blaschke products.

Of course, the points α = β1, β2, · · ·, βn depend on α, so we might write

them as α = β1(α), β2(α), · · ·, βn(α).



B(α) = B(β) and B′(β) = 0,




Theorem: (Cowen, 1974) (Ritt, 1922, ’23)

For B a finite Blaschke product, the branches of B−1 ◦B form a group

whose normal subgroups are associated with compositional

factorizations of B into compositions of two Blaschke products.

Recall the

Fundamental Lemma:




Recall the

Fundamental Lemma:




Use ideas about, W , the Riemann surface for B−1 ◦B to rewrite this as:

Fundamental Lemma(2):

Let B be a finite Blaschke product. Let F be the set

F = {α ∈ D : B(α) = B(β) for some β with B′(β) = 0}.

If S is a bounded operator on H2, then S is in {TB}′ if and only if

S∗Kα =

n∑j=1

cj(α)Kβj(α) for each α in D \ F .

We use this to write Sf as a function of α in the disk.

Theorem: (Cowen, 1978). Let B, F , and W be as above.

If S is a bounded operator on H2 that commutes with TB, then there is

a bounded analytic function G on the Riemann surface W so that for f

in H2,

(Sf )(α) = (B′(α))−1∑

G((β, α))β′(α)f (β(α)) (1)

where the sum is taken over the n branches of B−1 ◦B at α. Moreover,

if α0 is a zero of order m of B′, and ψ1, ψ2, · · ·, ψn is a basis for((B −B(α0))H

2)⊥

, then G has the property that∑G((β, α))β′(α)ψj(β(α)) has a zero of order m at α0 (2)

for j = 1, 2, · · · , n.

Conversely, if G is a bounded analytic function on W that has

properties (2) at each zero of B′, then (1) defines a bounded linear

operator on H2 with S in {TB}′.

In 2006, Cowen and Gallardo-Gutierrez, in connection with their study of

adjoints of composition operators, developed a formal class of operators

called ‘multiple-valued weighted composition operators’. The operators S in

{TB}′ are just such operators.





In the past few years, Douglas, Sun, and Zheng, and Douglas, Putinar, and

Wang, and others have used related tools to study problems concerning

commutants of TB on the Bergman space, such as consideration of the

reducing subspaces of TB.





In the past few years, Douglas, Sun, and Zheng, and Douglas, Putinar, and

Wang, and others have used related tools to study problems concerning

commutants of TB on the Bergman space, such as consideration of the

reducing subspaces of TB.

Observation:

The class of ‘multiple-valued weighted composition operators’, an extension

of classes of algebras of operators generated by multiplication and

composition operators, appear to be useful in the study of certain kinds of

problems in operator theory, including questions related to commutants.

Theorem: (C. & Wahl, 2012). Let B, F , and W be as above.

If S is a bounded operator on A2 that commutes with TB, then there is

a bounded analytic function G on the Riemann surface W so that for f

in A2,

(Sf )(α) = (B′(α))−1∑

G((β, α))β′(α)f (β(α)) (3)

where the sum is taken over the n branches of B−1 ◦B at α. Moreover,

if α0 is a zero of order m of B′, and ψ1, ψ2, · · ·, ψn is a basis for((B −B(α0))A

2)⊥

, then G has the property that∑G((β, α))β′(α)ψj(β(α)) has a zero of order m at α0 (4)

for j = 1, 2, · · · , n.

Conversely, if G is a bounded analytic function on W that has

properties (4) at each zero of B′, then (3) defines a bounded linear

operator on A2 with S in {TB}′.

Theorem: (Cowen, 1978).

If B is a finite Blaschke product and S is a bounded operator on H2

such that STB = TBS,

then for all f in H∞, Sf is also in H∞.

Theorem: (Cowen, 1978).

If B is a finite Blaschke product and S is a bounded operator on H2



Theorem: (C. & Wahl, 2012).

If B is a finite Blaschke product and S is a bounded operator on A2







Ideas of the proof:

(Sf )(α) = (B′(α))−1∑

G((β, α))β′(α)f (β(α)) (3)

First, f is assumed to be bounded on the disk:

so |f (β(α))| is bounded by ‖f‖∞.





Ideas of the proof:

(Sf )(α) = (B′(α))−1∑

G((β, α))β′(α)f (β(α)) (3)

Second, B is an n - to - 1 map of the Riemann sphere to itself, so it has n

poles outside the closed unit disk. In particular, B is analytic in a disk

strictly larger than D and the β′(α) are bounded in a disk larger that D.





Ideas of the proof:

(Sf )(α) = (B′(α))−1∑

G((β, α))β′(α)f (β(α)) (3)

Third, B′ has 2n− 2 zeros on the Riemann sphere, n− 1 in D and the

other n− 1 are reflections of these outside the closed unit disk.

In particular, B′ is analytic and non-zero in an annulus strictly containing

the unit circle.

This means (B′(α))−1 is bounded near the unit circle.





Ideas of the proof:

(Sf )(α) = (B′(α))−1∑

G((β, α))β′(α)f (β(α)) (3)

Finally, the sum appears to depend on all n of the branches of B−1 ◦B

simultaneously.

Of course, it does, but using bounded analytic functions as multipliers, we

can eliminate all but one term in the sum (3).

This allows us to show that each term of the sum G((β, α)) is bounded

separately and there are n bounded terms in the sum.





Corollary:

The commutants of TB as an operator on H2 and of TB as an operator

on A2 are ‘the same’.





Corollary:



The bounded analytic functions on the disk are dense in both H2 and A2.

Since these functions are mapped in the same way as vectors in H2 and A2,

the operators agree on all vectors common to H2 and A2.

These ideas apply in the same way to the weighted Bergman spaces as well.





Corollary:



Corollary:



is a finite Blaschke product, there is finite Blaschke product B with

{Tψ}′ = {TB}′ as operators on A2





Corollary:



Corollary:

If P is a bounded operator acting on H2 such that P 2 = P and

TBP = PTB, then P is a bounded an operator acting on A2 such that

P 2 = P and TBP = PTB.

The result

Corollary:

If P is a bounded operator acting on H2 such that P 2 = P and

TBP = PTB, then P is a bounded an operator acting on A2 such that

P 2 = P and TBP = PTB.

leads to some obvious, but still unsolved problems: “Which of the

projections that commute with TB on the Bergman space are self-adjoint?”

It is easy to see that many more self-adjoint projections commute with TB

on H2 than on A2 because multiplication by B is an isometry in H2, but

not on A2.

The question “What is {TB, T∗B}′ ?” is largely unstudied!

Thank You!

References

[1] A. Beurling, On two problems concerning linear transformations in Hilbert space,Acta Math. 81(1949), 239–255.

[2] C.C. Cowen, The commutant of an analytic Toeplitz operator,Trans. Amer. Math. Soc. 239(1978), 1–31.

[3] C.C. Cowen and E.A. Gallardo-Gutierrez, A new class of operators and a descriptionof adjoints of composition operators, J. Functional Analysis 238(2006), 447–462.

[4] J.A. Deddens and T.K. Wong, The commutant of analytic Toeplitz operators,Trans. Amer. Math. Soc. 184(1973), 261–273.

[5] R.G. Douglas, M. Putinar, and K. Wang, Reducing subspaces for analytic multipliersof the Bergman space, preprint, 2011.

[6] R.G. Douglas, S. Sun, and D. Zheng, Multiplication operators on the Bergman spacevia analytic continuation, Advances in Math. 226(2011), 541–583.

[7] J.F. Ritt, Prime and composite polynomials,Trans. Amer. Math. Soc. 23(1922), 51–66.

[8] J.F. Ritt, Permutable rational functions,Trans. Amer. Math. Soc. 25(1923), 399–448.

[9] J.E. Thomson, Intersections of commutants of analytic Toeplitz operators,Proc. Amer. Math. Soc. 52(1975), 305–310.

[10] J.E. Thomson, The commutant of a class of analytic Toeplitz operators,Amer. J. Math. 99(1977), 522–529.

Slides available: http://www.math.iupui.edu/˜ccowen

Blank Page

If B(z) = z2(z − .51− .5z

)2

, the group B−1 ◦B is isomorphic to D4.

D4 has several normal subgroups, and most give trivial factorizations of B

into the composition of a Blaschke product of order 1 and one of order 4.

However, there is a normal subgroup that “finds” the non-trivial

decomposition of B as B = J1 ◦ J2 where J1(z) = z2 and J2(z) = zz − .51− .5z

commutants of finite blaschke product multiplication ...ccowen/talks/wabash1405.pdfcommutants of...

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